Why do vertices move when 'Smooth Vertex' is applied to mesh?
when exporting to FBX, Blender looks at the Modifier Stack of each object and applies the RENDER modifiers to the object. I can only guess without checking the file, but I suspect that you have a SubSurf modifier on your geometry.

Uncheck the Render Visibility (red arrow is pointing at it in the screenshot) and export again, it should be fine then.